424 CURRENTS AND MAGNETIC SHELLS. 



point about the current in the direction of the hands of a watch. 

 Let a denote the angle PO/. For a very small displacement PP' 

 in the direction of the force, the work on a positive mass equal 

 to unity will be 



dT = < x PP' = <$>ada = kida. 



Fig. 96. 



As the angle /5, which the right line PO makes with a parallel 

 P#' to the axis, is complementary to a, we may write 



</T = -kid ft. 



This work is equal to the corresponding fall of potential - dV ; 

 we deduce therefrom 



and therefore 



V = fa'j3 + const. 



We may observe that the angle /5 is the rectilinear angle of the 

 dihedral angle of two planes drawn from the point P, one in the 

 current, the other parallel to the current and to the jc-axis ; twice this 

 angle /3 measures the surface o> of the spindle, which the dihedron 

 cuts through a sphere of unit radius with its centre at P ; we may 

 accordingly write 



ki 

 V = w + const. 



2 



The surface o> is evidently the solid angle under which the plane of 

 xz is seen from the point P, unlimited in one direction, and 

 bounded in the other by the current that is to say, the apparent 

 surface of this plane. 



